Questions: The area of a rectangle is 70 ft², and the length of the rectangle is 11 ft less than three times the width. Find the dimensions of the rectangle.

The area of a rectangle is 70 ft², and the length of the rectangle is 11 ft less than three times the width. Find the dimensions of the rectangle.

Transcript text: The area of a rectangle is $70 \mathrm{ft}^{2}$, and the length of the rectangle is 11 ft less than three times the width. Find the dimensions of the rectangle.

Solution

Solution Steps

Step 1: Start with the given area equation $A = L \cdot W$.

Step 2: Substitute the linear relationship $L = k \cdot W + m$ into the area equation to get $A = (k \cdot W + m) \cdot W$.

Step 3: Rearrange the equation to form a quadratic equation in terms of $W$: $k \cdot W^2 + m \cdot W - A = 0$.

Step 4: Solve the quadratic equation for $W$ using the quadratic formula: $W = \frac{-m \pm \sqrt{m^2 + 4 \cdot k \cdot A}}{2 \cdot k}$.

Step 5: Once $W$ is found, use the linear relationship $L = k \cdot W + m$ to find $L$.

Step 6: Check both solutions for $W$ (if applicable) to ensure they make sense in the context of the problem (e.g., positive dimensions).

Final Answer: One possible set of dimensions for the rectangle are Width = 7 and Length = 10.

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